Convex Hulls and Polygons in Computer Science

1. Polygons and the convex hull Prof. Noah Snavely CS1114 http://www.cs.cornell.edu/courses/cs1114

2. Administrivia • Assignment 3 due this Friday by 5pm • Please sign up for slots on CMS

3. Finding the lightstick center • Threshold the image • Find blobs (connected components) • Find the largest blob B • Compute the median vector of B

4. Finding the lightstick center • But we also want to control the robot based on the orientation of the lightstick

5. Finding the lightstick center • So far we’ve only built functions that take a set of points and return another point • With one exception… • How can we express the shape of the lightstick?

6. Finding the lightstick center • We’ll try to come up with a simple polygon to describe the lightstick • Simplest polygon:the bounding box

7. Bounding box • Not as informative as we might like • Let’s come up with a polygon that fits better…

8. Detour: convex polygons • A polygon P is convex if, for any two points A, B inside P, all points on a line connecting A and B are also inside P B A

9. Convex polygons • Which polygons are convex? 2 3 1 4

10. Testing convexity • How can we test if a polygon P is convex? • Consider the smallest convex polygon containing P • Called the CONVEX HULL • What is the convex hull if P is convex?

11. Convex hull • Can also define for sets of 2D points: the smallest convex shape containing a set of 2D points from http://en.wikipedia.org/wiki/Convex_hull

12. Convex hull of point sets • We can use this to find a simple description of the lightstick’s shape • How can we compute the convex hull?

13. Computing convex hulls • Idea: two points are an edge in the convex hull if ___________________

14. Computing convex hull • Which two horizontal lines touch points on the convex hull? • Which two vertical lines? •  It is easy to identify at least four points that are part of the convex hull

15. Gift-wrapping algorithm • Start at lowest point • Rotate the line until we hit another point • All other points will lie on one side of this line • Look for the point that gives you the largest angle with the current line • Repeat • You’re done when you get back to the starting point Figure credit: Craig Gotsman

16. The details... } • Start at lowest point • Rotate the line until we hit another point • All other points will lie on one side of this line • Look for the point that gives you the largest angle with the current line • Repeat • You’re done when you get back to the starting point • How do we implement this part?

17. Vectors • To do this, let’s talk about 2D vectors • A vector v = (x, y) is an “arrow” with a direction and length • Similar to a 2D point y-axis (x, y) v x-axis (0,0)

18. Vectors length of v : ||v|| ||v|| = direction of v: (x, y) v (0,0)

19. Vectors v + u = (vx + ux, vy + uy) v u (0,0)

20. Vectors v v - u = (vx-ux, vy-uy) u (0,0)

21. Vectors • Can also “multiply” two vectors: • Dot product: • Useful fact:

22. Back to the convex hull w Which point is next? Answer: the point w that maximizes the angle between u – v and w – v v u - v u What is the running time of this algorithm?

23. Lightstick orientation • We have a convex shape • Now what? • Want to find which way it’s pointed • For now, we’ll find the two points that are furthest away from each other, and call that the “major axis”

24. Questions?

25. Computing the convex hull • Gift wrapping algorithm (“Jarvis march”) • Quickhull • (divide and conquer)